Conductor of an abelian variety

Last updated July 08, 2020

In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is. It is connected to the ramification in the field generated by the torsion points.

Definition

For an abelian variety A defined over a field F as above, with ring of integers R, consider the Néron model of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme over

Spec(R)

(cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism

Spec(F) → Spec(R)

gives back A. Let A⁰ denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a maximal ideal P of R with residue field k, A⁰_k is a group variety over k, hence an extension of an abelian variety by a linear group. This linear group is an extension of a torus by a unipotent group. Let u_P be the dimension of the unipotent group and t_P the dimension of the torus. The order of the conductor at P is

f_{P}=2u_{P}+t_{P}+\delta _{P},\,

where $\delta _{P}\in \mathbb {N}$ is a measure of wild ramification. When F is a number field, the conductor ideal of A is given by

f=\prod _{P}P^{f_{P}}.

Properties

A has good reduction at P if and only if $u_{P}=t_{P}=0$ (which implies $f_{P}=\delta _{P}=0$ ).
A has semistable reduction if and only if $u_{P}=0$ (then again $\delta _{P}=0$ ).
If A acquires semistable reduction over a Galois extension of F of degree prime to p, the residue characteristic at P, then δ_P = 0.
If $p>2d+1$ , where d is the dimension of A, then $\delta _{P}=0$ .
If $p\leq 2d+1$ and F is a finite extension of $\mathbb {Q} _{p}$ of ramification degree $e(F/\mathbb {Q} _{p})$ , there is an upper bound expressed in terms of the function $L_{p}(n)$ , which is defined as follows:

Write

n=\sum _{k\geq 0}c_{k}p^{k}

with

0\leq c_{k}<p

and set

L_{p}(n)=\sum _{k\geq 0}kc_{k}p^{k}

. Then^[1]

(*)\qquad f_{P}\leq 2d+e(F/\mathbb {Q} _{p})\left(p\left\lfloor {\frac {2d}{p-1}}\right\rfloor +(p-1)L_{p}\left(\left\lfloor {\frac {2d}{p-1}}\right\rfloor \right)\right).

Further, for every

d,p,e

with

p\leq 2d+1

there is a field

F/\mathbb {Q} _{p}

with

e(F/\mathbb {Q} _{p})=e

and an abelian variety

A/F

of dimension

d

so that

(*)

is an equality.

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References

↑ Brumer, Armand; Kramer, Kenneth (1994). "The conductor of an abelian variety". Compositio Math. 92 (2): 227-248.

S. Lang (1997). Survey of Diophantine geometry . Springer-Verlag. pp. 70–71. ISBN 3-540-61223-8.
J.-P. Serre; J. Tate (1968). "Good reduction of Abelian varieties". Ann. Math. The Annals of Mathematics, Vol. 88, No. 3. 88 (3): 492–517. doi:10.2307/1970722. JSTOR 1970722.

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[1] Brumer, Armand; Kramer, Kenneth (1994). "The conductor of an abelian variety". Compositio Math. 92 (2): 227-248.

Conductor of an abelian variety

Contents

Definition

Properties

Related Research Articles

References